indifference (there is no relation).
Many times before we said that the system does not affect the environment; therefore, the interaction between them can be considered either as commensalism or as amensalism or as indifference.
If we enumerate all (n + 1) species of the ecosystem (including the (n + 1)th quasi-species "environment"), the overall structure of pair-wise interactions both within the system and between the system and the environment can be described by an (n + 1) X (n + 1) sign-matrix of interactions, S, where each (i, j)th element (i — j) is equal to +, — or 0. The sign (or no sign) indicates the qualitative result of the impact of the jth species upon the ith one.
In order to "complete" the description we have to define the diagonal elements of S. For this we have to understand how the state of each species (in particular, its size) impacts its own self-growth. Here, there are three possibilities: the state of species can stimulate its growth, it can inhibit growth and at least it can produce a neutral reaction. These three reactions can be denoted by +, — and 0, respectively. For the environment we assume the neutral self-interaction. By the same token we have defined the diagonal elements of S.
It is obvious that the number of elements of the matrix S is equal to LS = (n + 1)2. Since each element can be in one of the three states (+, — ,0) then the total number of different matrices S is nS = 3(n+1) < 10n /2. For instance, if the number of species n = 10 then nS < 1050. This is a huge number! Matrix S can be considered as a single word of length LS written in a three-letter alphabet; then the specific information (per one species) contained in S is equal to IS = k(n + 1)ln 3 < 1. 1k(n + 1). Comparing this expression and the one for I1 = k ln n (the information contained in the list of species), we see that limn!1 (IS /I1) < limn!1 [n/(ln n)] = 1. This implies that if the number of species is growing then the quantity of information which is needed for a qualitative description of the trophic network increases faster than the species diversity.
9.2. Topology of trophic network and qualitative stability
The concept of "qualitative stability" is defined in intuitive rather than formal terms for a system of interacting compartments of any kind. This means that a system (or more correctly, its equilibrium) holds its stability under any of those quantitative variations in the strength of linkages between the compartments, as long as qualitatively the interactions in the system remain unchanged. It is clear that in this case the stability of matrix S is to be analysed.
The appeal that this concept has to an ecologist is obvious: while it is always hard to estimate quantitatively the strength of linkages, it is much easier to make a qualitative conclusion about the type of intra- and interspecies relationships for all species. Then from this qualitative knowledge, i.e. from the signs of interactions between each pair of species, it may be possible to speculate about the stability in the whole class of "qualitatively similar" ecosystems. The latter implies that all these ecosystems have the same Sign Directed Graph (SDG) that corresponds to the matrix S. As an example, let us consider two S-matrices and the corresponding SDGs (Figs. 9.1a,b and 9.2a,b). They describe trophic chains with length 5 that differ from one another only by the following. In the first chain, a self-regulated species belongs to the first trophic level while in the second chain it belongs to the third level.
We can say that the system is qualitatively stable if (Svirezhev and Logofet, 1978; Logofet, 1993)
1. There cannot be any self-stimulated species, and at least one species must be self-limited.
2. There are no relations of competition (--) and mutualism (+ + ).
3. There are no directed loops (or cycles) of length 3 or more in the system.
This is probably the most severe restriction for ecosystems. In particular, it excludes all "omnivory" cases where a predator feeds on two prey species, one of which is also food to another, or, in more general terms, when a predator feeds on more than one trophic level.
4. The system must contain some m "prey-predator" pairs (i.e. 2-cycles) such that the rest (n — 2m) species are self-regulated (i.e. 1-cycle).
Assume that we can extract from the overall graph several sub-graphs that contain only prey -predator relations and isolated vertices (so-called predation graphs). If all these predation substructures are qualitatively stable, then the system as a whole will also be qualitatively stable. In order to test this statement it is sufficient to study special topological properties of these substructures. This procedure is called a "colour test".
We colour all self-regulated vortices (for instance, black) keeping others uncoloured (white). We say that "a predation graph passes the colour test if there exist white vertices, any one of which are linked to at least another one white vertex; and if a black vertex is linked to a white one, then it is also linked to at least another one white vertex". If such colouring is impossible then the graph fails the test. In this latter case the predation community is qualitatively stable.
Let us consider from this point of view the trophic chains shown in Fig. 9.2a,b. Both of them can be described by predation graphs. In case (a) the colour test is failed since the first
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